Making wounds matter.

[Knaight]

I’m really not that concerned about the system or mechanics. I’m more concerned about the story aspect.

All typical games assume that the characters can heal up fast, and the story flows with that assumption.

Wounded characters wouldn’t be at 100% when it comes to their skills and abilities. A thief with a twisted ankle will find it hard to climb up the side of a building to get to that balcony.

That’s fine. I’m just trying to figure out how to adjust the challenge levels in the game to adapt to the condition of the characters

Generally you'd just tone down some of the more ridiculous parts of said challenge level (e.g. the combat frequency assumed in D&D games) and generally operate at a slower pace with more gaps in the action. It's actually pretty easy to get used to.

I'd actually recommend using FATE: accelerated.
It clears out a lot of the unneeded chaff that FATE brings along from FUDGE and puts the aspects (the part of FATE that is actually unique and worthwhile) in the forefront.

I'd go the other way - use Fudge instead of Fate, which puts back in several excellent mechanics that Fate removed. There's the simple workhorse mechanics of the more complex Fudge combat system (ODF, DDF, the wound track), more use of modifiers, and standout mechanics like Scale which let Fudge work across a wide range of scales better than almost any other game. It's also an easy transition from more conventional games, as Fudge is in many ways very conventional - just extremely well built.